IN   MEMORIAM 
FLORIAN  CAJORl 


leath's  Mathematical  Monographs 

Issued  under  the  general  editorship  of 

Webster  Wells,  S.  B. 

:i)fcssor  of  Mathematics  in  the  Massachusetts  Institute  of  Technology 


GRAPHS 


BY 


ROBERT  J.  ALEY,  A.M.,  Ph.D. 


Professor  or  Mathematics  in 
Indiana  University 


.  Heath  &  Co.,  Publishers 

New  York  Chicago 


Number  6 


Price,  Ten  Cents 


Heath's  Mathematical  Monographs 

Number  6 


GRAPHS 

ALEY 


Dr.  Aley 

has  prepared  a  valuable 

Chapter  on  Graphs 

for 

Wells's  Essentials  of  Algebra 

and 
Wells's  New  Higher  Algebra. 

The  editions  containing  this 
chapter  will  be  supplied 
when  specially  ordered. 


HEATH'S    MATHEMATICAL    MONOGRAPHS 
Number  6 


GRAPHS 


BY 


ROBERT  J.  ALEY,  A.M.,  Ph.D. 

■A 
PROFESSOR  OF  MATHEMATICS  IN 
INDIANA  UNIVERSITY 

T 


c- 


BOSTON,  U.S.A. 

D.  C.  HEATH  &  CO.,  PUBLISHERS 
1902 


(an  '     / 


Copyright,  1902, 
By  D.  C.  Heath  &  Co. 


CAJORI 


INTRODUCTORY   STATEMENT 

At  the  present  time  the  Graph  is  used  so  extensively 
in  many  lines  of  work  that  it  is  necessary  for  the  non- 
technical reader  to  know  something  of  it.  When  we 
note  that  the  fluctuations  in  the  price  of  wheat,  the 
changes  of  temperature  for  a  month,  the  age  of  conver- 
sion in  children,  the  advancement  in  learning  a  trade, 
the  strain  on  a  girder  under  different  loads,  the  death 
rate  at  different  ages,  and  the  solutions  of  numerical  and 
algebraical  problems  have  alike  been  subjected  to  graphi- 
cal methods,  we  conclude  that  elementary  mathematics 
should  take  some  note  of  the  subject. 

Graphical  methods  permeate  so  many  subjects  and 
may  be  used  so  freely  in  the  different  parts  of  elemen- 
tary mathematics  that  it  seems  that  there  may  be  use  for 
a  brief  treatment  outside  of  the  text-book.  Such  a  treat- 
ment in  the  hands  of  the  teacher  gives  him  the  power  to 
use  it  whenever  the  occasion  demands.  Such  a  treat- 
ment may  also  help  the  general  reader  to  an  understand- 
ing of  the  graphical  treatment  now  given  to  so  many 
subjects. 


GRAPHS. 

Definitions. 

A  Graph  is  a  representation  by  means  of  lines, 
straight  or  curved,  of  some  set  of  measured  or 
numerically  represented  facts. 

Axes.  Two  lines  intersecting  at  right  angles,  as 
in  Fig.  I,  are  called  the  Axes  of  Coordinates.  OXy 
the  horizontal  one,  is  the  /-axis,  or  Axis  of  Abscis- 


II 


I 

-l-ar,-f2/ 


—x.-y 
III 


+  a?,-2/ 
IV 


Y' 
Fig.  I. 


sas ;  O  Y,  the  vertical  one,  is  the  K-axis,  or  Axis  of 
Ordinates.  O,  the  intersection  of  the  axes,  is  the 
origin. 

Quadrants.  The  axes  divide  the  plane  into  four 
parts,  called  quadrants.  These  quadrants  are  num- 
bered from  I  to  IV,  as  in  Fig.  i. 

I 


2  Graphs. 

Coordinates.  A  po,int  is  located  when  its  distance 
,ujid  direcliori  fr6m  each  of  the  axes  is  known.  The 
distance  from  OV  is  the  ;r-distance,  or  abscissa. 
The  distance  from  OX  is  the  ^-distance,  or  ordinate. 
The  two  distances  constitute  the  Coordinates  of  the 
point.  A  point  is  denoted  by  the  symbol  (x,  }^), 
where  ;r  is  the  abscissa  and  j^  the  ordinate. 

Convention  as  to  Signs.  In  the  representation  of 
points,  distances  to  the  right  of  the  F-axis  are  posi- 


PJ-3.,3) 


P3(-3,-l) 


i 

0    ! 


Pi(M) 


KC  2.-3) 


Fig.  2. 

tive,  to  the  left  negative.  Distances  above  the 
JT-axis  are  positive,  those  below  negative.  An  x  in 
the  first  or  fourth  quadrant  is  + ,  in  the  second  or 
third  it  is  — .  A  _7  in  the  first  or  second  quadrant 
is  +,  in  the  third  or  fourth  it  is  — .  These  are 
indicated  in  Fig.  i. 

Plotting  Points.  To  locate  the  point  /*i(3,  4), 
we  measure  3  units  to  the  right  of  (9Fand  then 
measure  4  units  up  from  OX.  (See  Fig.  2.)  The 
point  -PaC—  2,  3)  is  2  units  to  the  left  oi  OV  and  3 


Temperature  Curve.  3 

units  above  OX.  The  point  Pjy—  3,  —  i)is  3  units 
to  the  left  of  (9Fand  i  unit  below  OX.  The  point 
P4(+  2,  —  3)  is  2  units  to  the  right  oi  OY  and  3 
units  below  OX. 

Temperature  Curve. 

The  temperature  at  noon  on  twenty  successive 
days  was  as  follows:  60,  62,  64,  63,  61,  6^,  73,  75, 
74,  72,  70,  68,  69,  70,  74,  70,  67,  65,  63,  64. 

To  show  this  graphically  we  take  as  our  X-axis 
(Fig.  3)  a  line  which  we  let  represent  a  temperature 
of  60  degrees.  Each  unit  along  this  line  represents 
one  day,  and  each  unit  above  the  line  one  degree 
of  heat.  The  temperature  on  the  first  day,  60 
Y 


Fig.  3- 

degrees,  is  at  the  origin,  O.  The  temperature  on 
the  second  day  is  62,  and  is  shown  at  the  point 
(i,  2);  that  is,  I  to  the  right  of  o  and  2  above  the 
X-axis.  The  temperature  of  the  other  18  days  is 
shown  in  a  similar  way.  If  a  smooth  curve  is 
drawn  through  these  20  points,  the  result  is  the 
temperature  curve,  or  the  graph  of  the  noon  tem- 
perature for  20  days. 


4  Graphs. 

In  the  same  way  the  graph  may  be  used  to  show 
the  fluctuations  in  the  price  of  wheat,  the  increase 
in  skill  in  learning  a  trade,  or  in  fine  anything  that 
may  be  represented  by  the  combination  of  two 
series  of  facts. 

Solution  of  Problems  by  Graphs. 

1.  A  travels  4  7niles  an  hour,  B  6  miles  an  hour. 
If  A  has  2  hours  the  start,  when  and  where  will  B 
overtake  hint  f 

In  this  problem  let  each  space  along  the  JT-axis 
(Fig.  4)  represent  i  mile,  and  each  space  along  the 
F-axis  represent  one-half  hour.  At  the  end  of 
the  first  hour  A  is  evidently  at  the  point  A.     At  the 


end  of  the  second  hour  he  is  at  B,  and  so  on.  His 
path  in  time  and  space  is  readily  seen  to  be  OP. 
B  does  not  start  until  2  hours  after  A  starts,  so  his 
path  begins  at  L,  four  spaces  (2  hours)  above  O. 
At  the  end  of  first  hour  B  is  at  C,  and  his  path  is 
LP.  B  overtakes  A  when  his  path  crosses  that  of 
A.    This  occurs  at  P.     A  perpendicular  from  P  to 


Solution  of  Problems  by  Graphs.  5 

OX  intersects  it  at  M,  24  spaces  to  the  right  of  O. 
B,  therefore,  overtakes  A  24  miles  from  starting- 
place.  The  length  of  the  perpendicular  PM  is  12 
spaces.  Hence  B  overtakes  A  6  hours  after  A 
starts,  or  4  hours  after  he  himself  starts. 

2.  Two  towns  P  and  Q  are  48  miles  apart.  A 
walks  from  Y  to  (^  at  the  rate  of  3  miles  an  Jiotir 
a7td  rides  back  at  12  miles  a7i  hour,  B  starts  from, 
Q  two  hours  after  A  starts  fvm  P,  a7td  rides  to  P 
at  the  rate  of  8  miles  an  hour,  and  walks  back  at  4 
miles  a7i  hour.  When  and  where  do  A  and  B  meet 
the  seco7id  time  ? 

Figure  5  shows  the  solution  to  this.  The  paths 
of  A  and  B  are  marked  and  can  be  easily  under- 


Fig-  5. 


stood  from  what  has  preceded.  Their  second 
meeting  place  is  M,  which  is  seen  to  be  36  miles 
from  P  and  12  miles  from  Q.     The  line  ML  is  17 


6  Graphs. 

spaces  long,  and  so  they  meet  17  hours  after  A 
starts,  each  vertical  space  here  having  been  chosen 
to  represent  one  hour. 

3.  A,  B,  and  C,  travelling  at  6,  8,  and  12  miles 
an  hour^  start  at  the  same  time  around  aji  island  48 
miles  in  circumference.  When  and  where  are  they 
again  all  together  ? 

The  graph  in  Fig.  6  shows  the  result  at  once. 
The  two  perpendiculars  OS  and  IR  are  48  spaces 
apart.  Since  the  road  is  a  circle,  any  point  in  IR 
simply  represents  the  completion  of  one  circuit  and 


Fig.  6. 

really  represents  a  new  starting-point  in  OS.  A's 
path  is  OPj  PQ,  QR,  three  complete  circuits.  B's 
path  is  OL,  LM,  MN,  NR,  four  complete  circuits. 
C's  path  is  OD,  DP,  PM,  MQ,  QT,  TS,  six  com- 


Solution  of  Problems  by   Graphs.  7 

plete  circuits.     At  the  end  of  these  circuits  they 
are  all  together.     The  time  IR  is  24  hours. 

4.  A  man  walking  from  a  town  A  to  another  B 
at  the  rate  of  4.  miles  aii  hour,  starts  one  hour  before 
a  coach  which  goes  12  miles  an  hour  and  is  picked 
up  by  tJie  coach.  O71  arriving  at  B,  he  observes  that 
his  coach  journey  lasted  2  hours.  Find  the  distance 
from  A  /^  B. 

Let  spaces  (Fig.  7)  to  the  right  be  miles  and 
spaces  up  be  quarter  hours.  AP  is  the  path  of  the 
man  while  walking.  The  carriage  path  is  CP.  The 
intersection  P  is  6  spaces  to  right  and  6  spaces  up. 
The  carriage  picks  the  man   up   6  miles  from  A 


Fig.  7. 


and  i\  hours  after  he  has  started.  The  destina- 
tion is  reached  2  hours  from  P  ;  that  is,  the  line  CP 
is  continued  until  it  cuts  a  time  line  8  spaces  above 
P  at  B.  AM  equals  30  miles  and  is  the  distance 
between  the  towns. 


Graphs. 


A  Linear  Equation  in  Two  Variables. 

3x-h4J^=  12. 

We  say  in  algebra  that  such  an  equation  is  inde- 
terminate, for  we  can  get  an  indefinite  number  of 
values  of  x  and  j^  that  will  satisfy  it.  To  get  a 
solution  we  need  only  to  assign  arbitrarily  \.o  x  3. 
value  and  then  solve  for  j.  The  following  is  a  set 
of  solutions  : 

x= o  y=  3 


X—   I 

7=2i 

X  —  2 

7=li 

^=3 

J=i 

;ir=4 

y^o 

X 


5 


7  =  - 


The  list  could  be  extended  indefinitely. 

We  can  write  these  solutions  as  the  points  (o,  3), 


Fig.  8. 


A  Linear  Equation  in  Two  Variables.      9 

(I,  2i),  (2,  1 1),  (3,  I),  (4,  o),  and  (5,  -  |)-  These 
points  may  be  plotted  as  in  Fig.  8.  It  will  now  be 
noticed  that  these  points  are  in  a  straight  Hne. 
The  line  is  called  the  graph  of  the  equation.  If 
the  line  be  produced  indefinitely  and  the  x  and  y 
of  any  point  found  by  measurement  from  the 
graph,  the  values  thus  found  will  satisfy  the  equa- 
tion. For  example,  if  we  select  the  point  Q,  we 
find  that  its  x,  OM,  is  -  4,  and  its  y,  MQ,  is  +  6. 
These  values  satisfy  the  equation  ^x  +  ^y—  12, 
for  3(- 4) +  4(6)=  12. 

An  equation  of  the  first  degree  in  two  variables 
always  has  a  straight  line  for  its  graph. 

ax  -{-  by  =  c  is  a  general  linear  equation  in  x 
and  y. 

A  set  of  solutions  is  as  follows : 


X  =  o  y 


x—  I  y 

x=  2  y 

^=3  y 


b 

_c  —  a 
b 

_c  —  2a 
~~~b 

b 
_  c  —  A,a 


etc.  etc. 


lO 


Graphs . 


We  see  that  a  change  of   i   in  the  value  of  x 


makes  a  change 


in  y.     If  these  points  were 


plotted,  they  would  appear  (Fig.  9)  very  much  like 
the  side  view  of  a  uniform  straight  stairway,   in 


Fig.  9. 


which  the  width  of  the  steps  is  i,  and  the  height 

-  •    The  points  are  readily  seen  to  be  in  a  straight 
0 

line. 


A  shorter  way  of  getting  the  graph. 

Since  the  linear  equation  always  represents  a 
straight  line,  we  can  draw  its  graph  if  we  know 
two  points  upon  it.  In  general,  the  two  points 
most  easily  determined  are  those  where  the  graph 
cuts  the  coordinate  axes.  The  point  on  the  Jf-axis 
is  found  by  putting  7  =  0  and  solving  for  x.  The 
point  on  the  F-axis  is  found  by  putting  x  —  o  and 
solving  for^. 


A   Linear  Equation  in  Two  Variables,      ii 
Example.     2x—  ^y—  lo. 
If  J  =  o,         .r  =  5  ; 

and  if  ;r  =  o,        y  —  —  2. 

Y 


(6.0) 


(0.-2) 


Fig.  10. 


The  require(i  graph  cuts  the  X-axis  at  (5,  o)  and 
the  F-axis  at  (o,  —  2).  Plotting  these  two  points, 
the  line  is  easily  drawn  as  in  Fig.  10. 


Simultaneous  linear  equations. 

(1)  r5;r  +  4;j/=22l 

(2)  \lx-\-    7=    9J 

Draw  the  graphs  of  these  two  equations  on  the 
same  diagram  as  in  Fig.  11.  It  is  found  that  the 
two  lines  intersect  at  a  point  P  whose  coordinates 
are  (2,  3).  The  x  and  y  (2  and  3)  of  this  point  is 
the  solution  of  the  equations. 

Two  simultaneous  linear  equations  in  x  and  y 
have  but  one  solution.     Each  equation  represents 


12 


Graphs. 


a  straight  Hne.     The  solution  is  the  point  common 
to  both  hnes ;  that  is,  the  intersection  of  the  hnes. 


Fig.  II. 

Two  straight  Hnes  can  only  intersect  in  one  point, 
so  there  is  but  one  solution. 


(1)  r    ^+    j=i 

(2)  L2.r  +  2^=  7 

If  we  undertake  to  solve  the  above  equations,  we 
encounter  a  difficulty.      We  find  that  we  cannot 

Y 


Fig.  12. 

eliminate  x  without  also  eUminating  y  at  the  same 
time. 


The  Quadratic  Equation.  13 

If  we  draw  the  graphs  of  these  equations,  we 
find  they  are  represented  as  in  Fig.  12.  The 
graphs  show  at  once  where  the  difficulty  is.  The 
Hues  are  parallel  and  so  do  not  intersect  at  all.  In 
the  language  of  mathematics,  they  intersect  at  in- 
finity, which  is  just  another  way  of  saying  that 
they  never  intersect. 

The  Quadratic  Equation. 

ax^-\-  bx  +  c  =y  is  an  equation  which,  whenj  =  o, 
is  the  type  form  of  the  quadratic  in  a  single  variable. 
If  the  quadratic  in  x  is  thought  of  in  the  above 
form,  it  readily  yields  to  graphical  representation. 

Graph  of  x'^  —  2  x  —  2,  =  o. 

We  write  x'^  —  2x  —  ■^=  y.  Solving  this  for  x 
in  the  usual  way,  we  get 


x=  I  ±  V4  -\-y. 

The  following  Hst  of   values  for  y  and  x  are 
readily  found : 

1.  y  =      o  x=  I     and  —  i 

2.  y  =  —  I  X  =  2.y  and  —    .7 
Z.  y  —  —  2  X—  2.4.  and  —  4 

4.  j/  =  — 3  X  =  2     and       o 

5.  J/  =  —  4  X  =  I      and       I 

6.  /  =      I  ,r  =  3.2  and  —  1.2 


H 


Graphs. 


7. 

y  = 

2 

X  =  3.4  and 

-  1-4 

8. 

y  = 

3 

;r  =  3.6  and 

-  1.6 

9. 

y=. 

4 

;r=  3.8  and 

-  1.8 

10. 

y^ 

5 

;r  =  4  and 

—  2 

11. 

etc 

12 

X  —  ^     and 
etc. 

-3 

Plotting  these  points  carefully  and  connecting 
them  by  a  smooth  curve,  we  get  the  result  shown 
in  Fig.  13.     It  is  seen  that  the  graph  in  this  case 


is  a  curve,  and  that  it  cuts  the  axis  of  X  in  two 
points.  These  points  are  at  distances  of  3  and 
—  I  from  the  origin.  3  and  —  i  are  the  two  roots 
of  the  quadratic  x'^ 


2X  —  ^  —  o. 


A  quadratic  always  represents  a  curve  that  can 
be  cut  in  two  places  by  one  straight  line. 


Write 


X^  —  2X  ■\-  I  =0. 


2x  •\-  \  —y. 


The  Quadratic  Equation. 


15 


Solving  for  x,  we 

J  have 

X  =   I 

±  Vj. 

^  =  0 

;r  =  I 

7=  I 

X  —  2  and 

0 

7  =  4 

;r  =  3  and 

—  I 

y  =  9 

;r  =  4  and 

-  2 

etc. 

etc. 

Plotting  these  points  and  drawing  a  smooth  curve 
through  them,  we  have  the  curve  shown  in  Fig.  14. 
This  curve  does  not  cross  the  axis  of  X,  but  touches 
it  at  the  point  (i,  o).  The  first  member  of  the 
given  equation  being  a  perfect  square,  the  equation 


Fig.  14. 


has  two  equal  roots.  The  graph  of  a  quadratic 
having  equal  roots  always  touches  the  axis  of  X  at 
a  distance  from  O  equal  to  one  of  the  equal  roots. 
If  we  consider  the  equation  x'^—6x-\-  10  =j/ 
and  treat  it  as  the  above,  we  get  a  graph  shown  in 


1 6  Graphs. 

Fig.  15.     This  curve  does  not  touch  the  axis  of  X. 
If  in  the  equation  x^  —  6x  -\   10  =  j/  we  put  7  =  0 


Fig.  15. 

and  solve,  we  get  imaginary  roots  for  x.  The  graph 
of  a  quadratic  having  imaginary  roots  does  not 
touch  the  axis  of  X. 

Simultaneous  Quadratics. 

1.  X  +y  =  2. 

2.  ;rj/  =  —  15. 

Square  (i),  subtract  4  times  (2),  and  extract  the 
square  root,  and  we  have 

{l)x-y=      8 
{^)  x-y  =  -^ 

In  Fig.  16  the  various  Hues  of  the  graph  are 
numbered  to  correspond  with  the  numbers  of  the 
equations. 


Simultaneous  Quadratics. 


17 


Equations  (i)  and  (2)  give  a  straight  line  and 
the  double-branched  curve  known  as  the  hyper- 
bola.    These  intersect  at  the  points  P,  Q,  whose 


0\ 


i^ 


< 


cs-r 


^^s-r- 


/ 


Fig.  16. 

coordinates  are  (-3,  5)  and  (5,  -  3).  These  are 
the  only  solutions  to  the  system  of  equations.  The 
auxiliary  lines  (3)  and  (4)  intersect  line  (i)  in  P 
and  Q,  and  hyperbola  (2)  in  R  and  5. 

\.    xy  =  12. 

2.  x^  -\-  y^  =  40. 

The  auxihary  equations  appearing  in  the  solution 
are : 

3.  jr-f  j/  =  +  8. 

4.  X  -^y  =  -^. 

5.  x—y  =  ^A,. 

6.  A'-j  =  -4. 


i8 


Graphs. 


The  graphs  of  all  these  equations  are  shown  in 
Fig.  17  by  the  corresponding  numbers. 

The  solutions  are  at  the  points  -P,  Q^  R,  and  5. 

Y 


Fig.  17. 


The  Complex  Number. 

In  making  general  the  treatment  of  quadratics, 
the  complex  number  becomes  necessary.  The 
imaginary  unit  or  z(  =  V—  i)  appears  in  the 
extraction  of  the  square  root  of  a  negative  quan- 
tity.     E.g.,     V  —  4  =  V4  (  —  I )  =  V4  •  V  —  I  = 

±  2  V—   I   =±22. 

a  +  bi  is  the  type  of  all  complex  numbers.  The 
imaginary  and  complex  numbers  are  graphically 
represented  by  means  of  Argand's  diagram. 

Two  axes  intersecting  at  right  angles  are  used 
just  as  in  ordinary  graphic  work.  The  horizontal 
one  is  the  axis  of  reals,  and  the  vertical  the  axis  of 
imaginaries. 


The  Complex  Number. 


19 


In  Fig.  18  CA  and  BD  intersect  at  right  angles 
at  O.  0A(—  i)  =  -  OA  =0C.  Hence  we  may 
regard  —  i  as  an  operator  which  reverses  OA,  or 


which  turns  it  about  O  through  an  angle  of  180°. 
We  might  then  think  of  2  =  V—  i  as  an  operator 
which  turns  OA  through  an  angle  of  90°,  or  OAz 
=  0B. 
Then  OAz^  =  OBi  ^0C\ 

OAi^=  OBi^=^  OCi  ^0D\ 
OAi^  =  OBP  =  Oa^=  ODi  =  OA. 

These  results  merely  show  that  when  we  regard 
i  as  an  operator  which  turns  a  quantity  through  an 
angle  of  90°,  we  get  results  consistent  with  the 
known  algebraic  set  of  facts  : 

•3  -^  _  V 


I 

\.i 


I 

i 

—  I 


I. 


20 


Graphs. 


To  represent  any  complex  number  as  ^  +  iy,  we 
measure  on  OX  a  distance  OM=x  and  a  perpen- 
dicular distance  PM  =  y.     The  point  P,  or  as  is 


iy 


Fig.  19. 

frequently  more  convenient,  the  line  OP,  is  said 
to  represent  x  +  iy.  OP  =  V,t'2  +  j/^  =  r,  radius 
vector,  and  when  taken  with  the  positive  sign  is 
called  the  modulus. 

The  angle  MOP  =  ^  is  called  the  amplitude. 

X  =  OP  cos  6  =^  r  cos  d ; 

J  =  (9/^  sin  ^  =  r  sin  ^. 

Hence  ;r  +  2^7  =  r(cos  ^  +  ^  sin  ^). 

If  OS  represents     x^  +  /;/j, 

and  6^7"  represents     x^  +  /^/g* 

then  OP  represents  {x^  +  iy^  +  (;r2  4-  ^3^2)' 

OP  being  the  diagonal  of  the  parallelogram  of 
which  OS  and  (97"  are  two  adjacent  sides. 


The  Complex  Number. 


21 


The  diagram  shows  at  once  that  if  OP  repre- 
"''"*"  ^  +  iy'  (say), 

then  y  =  jTj  +  x^  and  y  =  y\  +  ^2' 

and  hence  OP  =  OS  +  OT. 


Fig.  20. 

The  diagram  may  be  used  to  iUustrate  nearly  all 
the  principles  of  the  complex  number.  Enough 
has  been  given  to  show  its  adaptability. 


FACTORING 

AS   PRESENTED   IN 

WELLS'  ESSENTIALS  <?/ALGEBRA 

I.    Advanced  processes  are  not  presented  too  early. 
II.     Large  amount  of  practice  work. 

No  other  algebra  has  so  complete  and  well-graded  development  of 
this  important  subject,  presenting  the  more  difficult  principles  at  those 
stages  of  the  student's  progress  when  his  past  work  has  fully  prepared 
him  for  their  perfect  comprehension. 

The  Chapter  on  Factoring  contains  these  simple  processes  : 

CASE    I.      When  the  terms  of  the  expression  have  a  common  monomial  factor  — 
thirteen  examples. 
II.     When  the  expression  is  the  sum  of  two  binomials  which  have  a  common 
binomial  factor  —  twenty  examples. 

III.  When  the  expression  is  a  perfect  trinomial  square  —  twenty-six  examples. 

IV.  When  the  expression  is  the  difference  of  two  perfect  squares — fifty-five 

examples. 
V.     When  the  expression  is  a  trinomial  of  the  form  x^-{-ax-i-l? 

—  sixty-six  examples. 
VI.     When  the  expression  is  the  sum  or  difference  of  two  perfect  cubes  —  twenty 
examples. 
VII.      When  the  expression  is  the  sum  or  difference  of  two  equal  odd  powers  of 
two  quantities  —  thirteen  examples. 

Ninety-three  Miscellaneous  and  Review  Examples. 
Further  practice  in  the  application  of  these  principles  is  given  in  the  two  following 
chapters  —  Highest  Common  Factor  and  Lowest  Common  Multiple. 

In  the  discussion  of  Quadratic  Equations,  Solution  of  Equations  by 
Factoring  is  made  a  special  feature. 
Equations  of  the  forms  ;^*— 5.^  —  24  =  0,  2X^  —  x  =  o,x^-\-4x^  —  x  —  ^ 

—  o,  and  ;^J—  I  =0  are  discussed  and  illustrated  by  thirty  examples. 
The  factoring  of  trinomials  of  the  form  ax^+lx-\-c  and  ax'^+I^x^-^c, 

which  involves  so  large  a  use  of  radicals,  is  reserved  until  Chapter 

XXV,  where  it  receives  full  and  lucid  treatment. 

The   treatment   of  factoring  is   but   one  of  the  many  features  of 
superiority  in  Wells*  Essendals  of  Algebra. 

Ha/f  Leather,  338  pages.      Price  $1.10. 

D.     C.     HEATH     &     CO.,     Publishers 

BOSTON  NEWYORK  CHICAGO 


Wells's  Essentials  of  Algebra  is  the 
best  book  that  can  be  placed  in  the  hands  of 
a  preparatory  student.  The  one  essential  fea- 
ture that  I  admire  in  the  book  is  its  simplicity. 
So  many  authors  now  write  algebras  and  geom- 
etries to  see  how  hard  they  can  make  them, 
and  the  consequence  is  that  only  the  brightest 
students  acquire  any  real  knowledge  of  the 
subject. 

Prof.  Wells  leads  the  student  step  by  step, 
and  his  book  is  well  named  The  Essentials, 
as  it  contains  all  that  is  essential  to  a  student 
entering  any   of  our  great  universities. 

EDWIN  W.   RAND, 

Principal  Princeton  University  Academy. 


The  definitions ;  the  problems ;  the  chapter 
on  factoring  ;  the  factoring  of  trinomials  ;  the 
discussion  of  the  principles  of  fractions  ;  the  so- 
lution of  equations  by  factoring ;  all  combine  to 
make  Wells's  Essentials  of  Algebra  a  book  of  su- 
perior worth.      Such  a  book  I  can  recommend. 

E.  MILLER, 

Professor  of  Mathematics,   University  of  Kansas. 


Wells's 
Essentials  of  Geometry 

is  far  superior  in 

the  arrangement  of  propositions, 

the  choice  of  proofs,   and  in 

the  method  of  presenting   the  subject. 

The  student  is  stimulated  to  help    himself,   to 
make  his  own    proofs,    to    gain  logical  power. 

L.  B.  MULLEN,  Ph.D., 

Dept.  of  Math.,  Central  High  School, 
Cleveland,  0. 


In  the  order   of  theorems, 

the  proof  of  corollaries, 

the  grading  of  the  original  exercises,  and 

the  opportunity  for   original  work, 

Wells's    Essentials  of    Geometry   is  notably 

superior. 

C.  A.  HAMILTON, 

Dept.  of  Math.,  Boys'  High  School, 
Brooklyn,  N.   T. 


The  Vital  Difference 

lies  in  this: 


Wells's  Essentials  Plane  and  Solid 
Geometry  is  characterized  by  the 
inductive  method  of  demonstration, 
and  always  requires  the  student  to  do 
for  himself  the  maximum  amount  of 
reasoning  and  thinking  of  which  he  is 
capable,  while  most  geometries  require 
of  the  pupil  in  his  demonstrations  little 
personal  power  except  that  of  memo- 
rizing. They  give  the  reasons  for 
statements  ;  while  Wells,  whenever  the 
student  should  be  able  to  give  the 
reason  for  himself,  always  asks  why 
the  statement  is  true.  No  other  geom- 
etry develops  so  strongly  the  power 
of  vigorous  independent  thought. 


D.  C.  HEATH   &   CO.,   Publishers 

BOSTON  NEW     YORK  CHICAGO 


Wells's   Mathematical   Series. 

ALGEBRA. 
Wells's  Essentials  of  Algebra      .....     $i.xo 

A  new  Algebra  for  secondary  schools.  The  method  of  presenting  the  fundamen- 
tal topics  is  more  logical  than  that  usually  followed.  The  superiority  of  the 
book  also  appears  in  its  definitions,  in  the  demonstrations  and  proofs  of  gen- 
eral laws,  in  the  arrangement  of  topics,  and  in  its  abundance  of  examples. 

Wells's  New  Higher  Algebra       .....       1.3a 

The  first  part  of  this  book  is  identical  with  the  author's  Essentials  of  Algebra. 
To  this  there  are  added  chapters  upon  advanced  topics  adequate  in  scope  and 
difficulty  to  meet  the  maximum  requirement  in  elementary  algebra. 

Wells's  Academic  Algebra  .....      1.08 

This  popular  Algebra  contains  an  abundance  of  carefully  selected  problems. 

Wells's  Higher  Algebra    ......       1.32 

The  first  half  of  this  book  is  identical  with  the  corresponding  pages  of  the  Aca- 
demic Algebra.    The  latter  half  treats  more  advanced  topics. 

Wells's  College  Algebra   ......       1.50 

A  modem  text-book  for  colleges  and  scientific  schools.  The  latter  half  of  this 
book,  beginning  with  the  discussion  of  Quadratic  Equations,  is  also  bound  sep- 
arately, and  is  known  as  Wells's  College  Algebra,  Part  II.     $1.33. 

Wells's  University  Algebra  .  .  •  •  •      1.3a 

GEOMETRY. 

Wells's  Essentials  of  Geometry — Plane,  75  cts.;  Solid,  75  cts.; 

Plane  and  Solid    .  .  .  .  .  .  .1.25 

This  new  text  offers  a  practical  combination  of  moxe  desirable  qualities  than 
any  other  Geometry  ever  published. 

Wells's  Stereoscopic  Views  of  Solid  Geometry  Figures        •        .60 

Ninety-six  cards  in  manila  case. 

Wells's  Elements  of  Geometry  —  Revised  1894.  —  Plane,  75  cts.; 

Solid,  75  cts.;   Plane  and  Solid     .  .  .  .  .1.25 

TRIGONOMETRY. 
Wells's  New  Plane  and  Spherical  Trigonometry  (1896)         .     $1.00 

For  colleges  and  technical  schools.    With  Wells's  New  Six-Place  Tables,  $1.25. 

Wells's  Plane  Trigonometry        .  .  .  .  •        '75 

An  elementary  work  for  secondary  schools.     Contains  Four-Place  Tibles. 

Wells's  Complete  Trigonometry  .  .  .  .         .90 

Plane  and  Spherical.  The  chapters  on  plane  Trigonometry  are  identical  with 
those  of  the  book  described  above.     With  Tables,  $1.08. 

Wells's  New  Six-Place  Logarithmic  Tables     .  ,  .        .60 

The  handsomest  tables  in  print.     Large  Page. 

Wells's  Four-Place  Tables  .  .  •  •  •        .25 

ARITHMETIC. 
Wells's  Academic  Arithmetic      .....     $i.oo 

Correspondence  regarding  terms  for  introduction 

and  exchange  is  cordially  invited. 

D.  C.  Heath  &  Co.,  Publishers,  Boston,  New  York,  Chicago 


HEATSi  S  MATHEMATICAL  MONOGRAPHS 


ISSUED   UNDER  THE  GENERAL  EDITORSHIP  OF 

WEBSTER  WELLS,   S.B. 

Professor  of  Mathematics  in  the  Massachusetts  Institute  of  Technology 


It  is  the  purpose  of  this  series  to  make  direct  contribu- 
tion to  the  resources  of  teachers  of  mathematics,  by  pre- 
senting freshly  written  and  interesting  monographs  upon 
the  history,  theory,  subject-matter,  and  methods  of  teach- 
ing both  elementary  and  advanced  topics.  The  first  five 
numbers  are  as  follows :  — 

1.  FAMOUS  GEOMETRICAL  THEOREMS  AND  PROBLEMS  AND 

THEIR  HISTORY.     By  William  W.  Rupert,  C.E. 
i.  The  Greek  Geometers,    ii.  The  Pythagorean  Proposition. 

2.  FAMOUS  GEOMETRICAL  THEOREMS.     By  William  W.  Rupert. 

ii.  The  Pythagorean  Proposition  (concluded),    iii.  Squaring  the  Circle. 

5.  FAMOUS  GEOMETRICAL  THEOREMS.     By  William  W.  Rupert. 

iv.  Trisection  of  an  Angle,    v.  The  Area  of  a  Triangle  in  Terms  of 
its  Sides. 

4.   FAMOUS  GEOMETRICAL  THEOREMS.     By  William  W.  Rupert. 

vi.  The  Duplication  of  the  Cube.    vii.  Mathematical  Inscription  upon 
the  Tombstone  of  Ludolph  van  Ceulen. 

6.  ON  TEACHING  GEOMETRY.     By  Florence  Milner. 

Others  in  preparation. 

PRICE,    10  CENT5   EACH 


!;t;f!!!;! 


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BOSTON        NSW  TORE      CHICAOO 


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Gaylord  Bros.,  Inc 

Makers 

Stockton,  Calif. 

PAT.  JAN.  21,  1908 


77 


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